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Weak Drifts of Infinitely Divisible Distributions and Their Applications

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Abstract

Weak drift of an infinitely divisible distribution μ on ℝd is defined by analogy with weak mean; properties and applications of weak drift are given. When μ has no Gaussian part, the weak drift of μ equals the minus of the weak mean of the inversion μ′ of μ. Applying the concepts of having weak drift 0 and of having weak drift 0 absolutely, the ranges, the absolute ranges, and the limit of the ranges of iterations are described for some stochastic integral mappings. For Lévy processes, the concepts of weak mean and weak drift are helpful in giving necessary and sufficient conditions for the weak law of large numbers and for the weak version of Shtatland’s theorem on the behavior near t=0; those conditions are obtained from each other through inversion.

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Acknowledgements

The authors thank an anonymous referee for giving them valuable advice on improvement of the paper.

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Authors and Affiliations

  1. Hachiman-yama 1101-5-103, Tenpaku-ku, Nagoya, 468-0074, Japan

    Ken-iti Sato

  2. Department of Mathematics, Keio University, 3-14-1, Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan

    Yohei Ueda

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  1. Ken-iti Sato
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  2. Yohei Ueda
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Correspondence to Ken-iti Sato.

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Sato, Ki., Ueda, Y. Weak Drifts of Infinitely Divisible Distributions and Their Applications. J Theor Probab 26, 885–898 (2013). https://doi.org/10.1007/s10959-012-0419-2

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  • DOI: https://doi.org/10.1007/s10959-012-0419-2

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